Question

a) lim (2x+ \(\sqrt{4x^2-x+1}\)) 

x-> -∞

b) lim x\(\left(\sqrt{4x^2+1}-x\right)\)

x-> -∞

Answer 1

a/ \(=\lim\limits_{x\rightarrow-\infty}\dfrac{4x^2-4x^2+x-1}{2x-\sqrt{4x^2-x+1}}=\lim\limits_{x\rightarrow-\infty}\dfrac{\dfrac{x}{x}-\dfrac{1}{x}}{\dfrac{2x}{x}+\sqrt{\dfrac{4x^2}{x^2}-\dfrac{x}{x^2}+\dfrac{1}{x^2}}}=\dfrac{1}{2+2}=\dfrac{1}{4}\)

b/ \(=\lim\limits_{x\rightarrow-\infty}x^2\left(-\sqrt{\dfrac{4x^2}{x^2}+\dfrac{1}{x^2}}-\dfrac{x}{x}\right)=\lim\limits_{x\rightarrow-\infty}x^2.\left(-3\right)=-\infty\)